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American Journal of Physics

Quasiclassical quantization of the magnetic charge

P. Hraskó

Published Sep 1, 1977• 3 citations
Publisher

Abstract

Quasiclassical quantization of a spherical pendulum in which the moving point mass and the center have electric and magnetic charge, respectively, is discussed. It is shown that for this system quantum conditions do not simply select between classical orbits. The requirement of spherical symmetry leads to the quantization condition of Dirac for the product of the charges and the total angular momentum may be either an integer or a half-integer multiple of h/.

DOI
10.1119/1.11058

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Quasiclassical quantization of a spherical pendulum in which the moving point mass and the center have electric and magnetic charge, respectively, is discussed. It is shown that for this system quantum conditions do not simply select between classical orbits. The requirement of spherical symmetry leads to the quantization condition of Dirac for the product of the charges and the total angular momentum may be either an integer or a half-integer multiple of h/.
We study quantum—classical correspondence in terms of the coherent wave functions of a charged particle in two-dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than ħ is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 2π-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.
In the presence of a magnetic field, the Maslov quantization condition is not available in the original form. An alternative quantization condition is proposed with the aid of a principal U(1) bundle over a phase space and a connection whose curvature form is the charged symplectic form. By means of this quantization condition, quasiclassical eigenvalues of the Kepler problem with a centrifugal potential and Dirac’s monopole field are calculated, which turn out to coincide with the eigenvalues of the quantized problem.
We study quantum–classical correspondence in terms of the coherent wave functions of a charged particle in two- dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 2π-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.
One of the most important applications of the quantum mechanical formalism developed in the previous chapter is to angular momentum. This is because many microsystems are spherically symmetric, or approximately so, and angular momentum is therefore conserved. This chapter discusses that a careful study of quantized angular momentum leads to a completely new physical phenomenon – intrinsic spin. The fact that the angular momentum vector of a quantum particle must point in certain restricted directions is a bizarre quantum property. In some respects the abstract idea of an internal angular momentum is reminiscent of the classical concept of a spinning body. In classical mechanics angular momentum is added vectorially but in the quantum case the issue is more subtle. The procedure will be illustrated by the example of the addition of two spins. As in classical mechanics a charged quantum system with angular momentum possesses a magnetic moment μ.
The seminal paper by Niels Bohr followed by a paper by Arnold Sommerfeld led to a revolutionary Bohr–Sommerfeld theory of atomic spectra. We are interested in the information about the structure of quantum mechanics encoded in this theory. In particular, we want to extend Bohr–Sommerfeld theory to a full quantum theory of completely integrable Hamiltonian systems, which is compatible with geometric quantization. In the general case, we use geometric quantization to prove analogues of the Bohr–Sommerfeld quantization conditions for the prequantum operators Pf. If a prequantum operator Pf satisfies the Bohr–Sommerfeld conditions and if it restricts to a directly quantized operator Qf in the representation corresponding to the polarization F, then Qf also satisfies the Bohr–Sommerfeld conditions. The proof that the quantum spherical pendulum is a quantum system of the type we are looking for requires a new treatment of the classical action functions and their properties. For the sake of completeness we have provided an extensive presentation of the classical spherical pendulum. In our approach to Bohr–Sommerfeld theory, which we call Bohr–Sommerfeld–Heisenberg quantization, we define shifting operators that provide transitions between different quantum states. Moreover, we relate these shifting operators to quantization of functions on the phase space of the theory. We use Bohr–Sommerfeld–Heisenberg theory to study the properties of the quantum spherical pendulum, in particular, the boundary conditions for the shifting operators and quantum monodromy.